Primary hyperthyroidism (PHPT) is a condition in which increased levels of parathyroid hormone lead to high blood calcium. This may result in kidney stones and bone loss,. The only available cure is surgery to remove the involved parathyroid gland(s). Currently, there is no specific medical treatment. In postmenopausal women with PHP, estrogen therapy can reduce blood calcium levels, but only by a small amount. The purpose of this study is to evaluate the effectiveness of another drug, alendronate, in lowering blood calcium concentration and improving bone density in patients with primary hyperparathyroidism. Alendronate has been studied extensively in postmenopausal women who have osteoporosis and has been shown to significantly increase bone density and reduce fracture risk. In addition, alendronate decreases blood calcium in some patients with primary hyperparathyroidism. This will be a randomized placebo controlled double blind clinical trial. Approximately 30 patients with primary hyperparathyroidism will receive either alendronate 10 mg/day or placebo. These treatment groups will continue for 12 months and then be followed by an additional 12 months period in which the group who received placebo will then receive alendronate 10 mg/day. Patients initially receiving Alendronate will continue on the same dose. We will follow patients to see if their blood calcium returns to (or toward) normal levels. We will also follow them for any improvement in bone mineral density.